A numerical study of some questions in vortex rings theory
A numerical study on Neumann-Neumann methods for hp approximations on geometrically refined boundary layer meshes II. Three-dimensional problems
In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal.24 (2004)...
A Nyström method for boundary integral equations in domains with corners.
A parallel AMG for overlapping and non-overlapping domain decomposition.
A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods.
A parallel multigrid method using the full domain partition.
A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations
A parameter-free stabilized finite element method for scalar advection-diffusion problems
We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming...
A penalty method for approximations of the stationary power-law Stokes problem.
A Penalty Method for the Approximate Solution of Stationary Maxwell Equations.
A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation
We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization...
A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems
A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation are presented and analyzed theoretically. The positive number is supposed to be much less than the discretization step and the values of . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.
A piecewise P2-nonconforming quadrilateral finite element
We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...
A posteriori error analysis for parabolic variational inequalities
Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual...
A posteriori error analysis of the fully discretized time-dependent Stokes equations
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
A posteriori error analysis of the fully discretized time-dependent Stokes equations
The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
A posteriori error estimate of approximate solutions to a mildly nonlinear elliptic boundary value problem
A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems
We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.
A posteriori error estimates for a nonconforming finite element discretization of the heat equation
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...
A posteriori error estimates for a nonconforming finite element discretization of the heat equation
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main...